Abstract:The problem of supplying an analogue of a manifold whose sheaf of functions contains anticommuting elements has been approached in two ways. Either one extends the sheaf of functions formally, as in the category of graded manifolds , , or one mimicks the usual definition of a manifold, having replaced Euclidean space with a suitable product of the odd and even parts of an exterior algebra as in the category of supermanifolds . This paper establishes the equivalence of the category of supermanifolds with the category of graded manifolds.
- Marjorie Batchelor, The structure of supermanifolds, Trans. Amer. Math. Soc. 253 (1979), 329–338. MR 536951, DOI 10.1090/S0002-9947-1979-0536951-0
- F. A. Berezin, The method of second quantization, Pure and Applied Physics, Vol. 24, Academic Press, New York-London, 1966. Translated from the Russian by Nobumichi Mugibayashi and Alan Jeffrey. MR 0208930
- F. A. Berezin and G. I. Kac, Lie groups with commuting and anticommuting parameters, Mat. Sb. (N.S.) 82 (124) (1970), 343–359 (Russian). MR 0265520 F. Berezin and D. Leites, Supervarieties, Soviet Math. Dokl. 16 (1975), 1218-1222.
- L. Corwin, Y. Ne’eman, and S. Sternberg, Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry), Rev. Modern Phys. 47 (1975), 573–603. MR 0438925, DOI 10.1103/RevModPhys.47.573 B. de Witt, Differential supergeometry (in preparation).
- V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. MR 486011, DOI 10.1016/0001-8708(77)90017-2
- Bertram Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975) Lecture Notes in Math., Vol. 570, Springer, Berlin, 1977, pp. 177–306. MR 0580292
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
- Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485
- François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 258 (1980), 257-270
- MSC: Primary 58A05; Secondary 81C99
- DOI: https://doi.org/10.1090/S0002-9947-1980-0554332-9
- MathSciNet review: 554332